Brake systems, whether for aircraft or land vehicles, function by applying a retarding torque to the braked wheels of the vehicle that is in a direction opposite to the rotational direction of the wheel as the vehicle moves across a road or runway surface or any other surface (hereafter referred to as a “ground surface” regardless of actual surface type). The actual braking force that decelerates the vehicle is a function of wheel slip, i.e. the difference between the translational velocity of the vehicle (relative to the wheel axis) and the corresponding translational velocity of the wheel at the contact point with the ground surface. This braking force is directed opposite to the vehicle velocity vector and results from sliding friction between the tire and the ground surface, since the contact point is really a patch with a finite area.
For the purposes of this disclosure, we shall call the wheel rotation retarding torque that is created by the brake mechanism and applied brake torque, Ta. We shall further call the vehicle decelerating or braking force the drag force, Fb, and the related road/tire torque the drag torque, Tb. We further define a normalized, dimensionless, instantaneous wheel slip, S, defined as:S=(V−ω*r)/V, where   (1)                 V=translational vehicle velocity        ω=wheel angular speed        r=effective radius of the braked wheel in question        
Note, “wheel angular speed” is the same as “wheel angular velocity” in the context we use these terms in the following and does not means wheel linear speed which is wheel angular speed multiplied by wheel radius r ((ω*r in equation (1)). The simple term “wheel speed” will mean wheel angular speed or wheel angular velocity in the following. When wheel linear speed is meant, it will be explicitly stated as such. Strictly speaking, velocity is a vector and speed is a scalar, but that usage rule is violated all the time in publications and general conversation.
Translational vehicle velocity V may be regarded as a scalar in the above equation (1) since we will assume straight line braking with no wheel sideslip in the remainder of this discussion. However, this assumption is for ease in understanding the principle of operation of the SMO-based ABS described herein and in no way limits the applicability of the improved ABS and method of implementing it to straight line braking.
The wheel drag or “braking” force, Fb, is linearly proportional to the normal force, N, acting on the wheel and the frictional ground surface coefficient, μ(S). Thus, we may calculate:Fb=N*μ(S)   (2) The corresponding wheel drag or “braking” torque, Tb, is obtained by multiplying Fb by the effective wheel radius, r. However the friction coefficient μ varies with slip S in a typical manner as depicted in FIG. 1a. There is a value of slip S that results in a peak value for the friction coefficient μ for any given vehicle, tire, and ground surface condition. This optimum value of slip S for maximizing wheel surface friction will be denoted by optimal slip S* in the following.
The primary ABS goal is to regulate applied brake torque Ta such that friction coefficient μ remains as close as possible to its peak value shown in FIG. 1A, thereby maximizing deceleration for a given applied brake torque Ta. ABS methods described in the prior art have generally involved instrumentation and/or computer algorithms for directly estimating the instantaneous slip S. It has been assumed in some of these methods that a single value for optimal (e.g. S*=0.13) is representative of all ground surface conditions or is sufficiently close to the true optimal slip S* that the braking performance is acceptable although sub-optimal. The ABS brake controller then simply regulates brake hydraulic pressure of a hydraulic braking system or electrical current applied to an electro-mechanical braking system so as to maintain an instantaneous slip value S.
For example, one prior art aircraft antilock brake systems evaluated by NASA Langley in the early 80's, described in NASA Technical Paper 1959 by John Tanner et al (dated February 1982), used an un-braked nose-wheel to obtain a vehicle velocity measurement and wheel angular speed ω from the braked wheels to compute an instantaneous slip value S by direct application of equation (1) assuming the un-braked nose wheel angular speed is the same as the vehicle velocity . The ABS controller compared the instantaneous slip value S to the assumed optimal slip S* and then modulated the brake pressure accordingly. Unfortunately, the nose wheel angular speed is only a very noisy measurement of the true aircraft velocity during the landing roll. There could be instances where the nose wheel has not fully touched down yet the ABS needs to be applied. Furthermore, the nose wheel could be worn or have low pressure so as to have a rounded radius.
In a more recent example, the sliding mode controller described by Unsal and Kachroo in “Sliding Mode Measurement Feedback Control for Antilock Braking Systems” (IEEE Transactions on Control Theory, March 1999, vol. 7, pp. 271-281), while based on braked wheel angular speed measurements, still assumed that the desired optimal slip, S*, was a fixed value and known in advance. Furthermore, even though the words “sliding mode” appeared in the title, Unsal and Kachroo really implemented a sliding mode “controller”, not a sliding mode “observer”. A controller attempts to force a system state variable(s) to achieve or track a given desired value(s) and drives the error between the actual state variable(s) and desired value(s) to zero. An observer merely attempts to estimate a system state variable(s) as accurately as possible given measurements related to that state variable(s).
Unsal and Kachroo computed their instantaneous slip S dynamically by solving a differential equation for slip S based upon a non-linear observation of the vehicle state (vehicle speed and wheel angular speed) with wheel angular speed as the sole state measurement. Their sliding mode controller attempted to drive S to an assumed know optimal value S*. The fundamental problem with such approaches is that optimal slip S* is not really a known constant but rather varies from under 0.1 to over 0.20, dependent on ground surface conditions, although most often in the range 0.11 to 0.15 (thus partially justifying the frequently used value of S*=0.13 mentioned earlier). Furthermore, the μ-S curve may vary dynamically during a given braking situation as ground surface conditions change and may even differ from one braked wheel to the next.
More sophisticated ABS methods attempt to estimate friction coefficient μ continuously and then control braking so as to dynamically track the actual peak of the μ-S curve. An example disclosed in U.S. Pat. No. 5,918,951 by Rudd employs an 8-state Kalman filter to directly estimate friction coefficient μ using as inputs pilot brake pressure aircraft velocity, and aircraft acceleration in addition to wheel angular speed on each brake wheel and, optionally, wheel brake torque as well. The aircraft velocity and acceleration measurements are assumed to be obtained from the aircraft electronics (an inertial navigation system and/or air data system for example). However, the mathematical complexity associated with such a mathematical estimation procedure might limit the ability of the '951 patent method to function properly in real-time braking situations.
Even if an accurate measurement of both vehicle velocity and wheel angular speed is available so that the instantaneous slip S can be measured, the optimal slip S* associated with the peak μ is really unknown in advance of brake application and may even vary along the ground surface. Rudd attempts to actually estimate S* and then command a pressure to the brake actuator that will continuously drive instantaneous slip S to optimal slip S*, but his approach is very complex mathematically involving an 8-state Kalman filter solution that must be repeated many, many times per second. Furthermore, while wheel angular speed sensors are cheap, the velocity and acceleration sensors required by Rudd in his preferred embodiment are usually associated with expensive inertial or air data systems.
What is needed is a simple method and system that operates in real-time to improve braking and that does not require complex mathematical calculations that are difficult to perform in real-time and/or expensive on-board sensors.